A321768 Consider the ternary tree of triples P(n, k) with n > 0 and 0 < k <= 3^(n-1), such that P(1, 1) = [3; 4; 5] and each triple t on some row branches to the triples A*t, B*t, C*t on the next row (with A = [1, -2, 2; 2, -1, 2; 2, -2, 3], B = [1, 2, 2; 2, 1, 2; 2, 2, 3] and C = [-1, 2, 2; -2, 1, 2; -2, 2, 3]); T(n, k) is the first component of P(n, k).

3, 5, 21, 15, 7, 55, 45, 39, 119, 77, 33, 65, 35, 9, 105, 91, 105, 297, 187, 95, 207, 117, 57, 377, 299, 217, 697, 459, 175, 319, 165, 51, 275, 209, 115, 403, 273, 85, 133, 63, 11, 171, 153, 203, 555, 345, 189, 429, 247, 155, 987, 777, 539, 1755, 1161, 429

Offset

1,1

Comments

The tree P runs uniquely through every primitive Pythagorean triple.

The ternary tree is built as follows:

- for any n and k such that n > 0 and 0 < k <= 3^(n-1):

- P(n, k) is a column vector,

- P(n+1, 3*k-2) = A * P(n, k),

- P(n+1, 3*k-1) = B * P(n, k),

- P(n+1, 3*k) = C * P(n, k).

All terms are odd.

Every primitive Pythagorean triple (a, b, c) can be characterized by a pair of parameters (i, j) such that:

- i > j > 0 and gcd(i, j) = 1 and i and j are of opposite parity,

- a = i^2 - j^2,

- b = 2 * i * j,

- c = i^2 + j^2,

- A321782(n, k) and A321783(n, k) respectively give the value of i and of j pertaining to (A321768(n, k), A321769(n, k), A321770(n, k)).

Every primitive Pythagorean triple (a, b, c) can also be characterized by a pair of parameters (u, v) such that:

- u > v > 0 and gcd(u, v) = 1 and u and v are odd,

- a = u * v,

- b = (u^2 - v^2) / 2,

- c = (u^2 + v^2) / 2,

- A321784(n, k) and A321785(n, k) respectively give the value of u and of v pertaining to (A321768(n, k), A321769(n, k), A321770(n, k)).

Links

Rémy Sigrist, Rows n = 1..9, flattened

Kevin Ryde, Trees of Primitive Pythagorean Triples, section UAD Tree "row-wise A leg".

Wikipedia, Tree of primitive Pythagorean triples

Index entries related to Pythagorean Triples

Formula

Empirically:

- T(n, 1) = 2*n + 1,

- T(n, (3^(n-1) + 1)/2) = A046727(n),

- T(n, 3^(n-1)) = 4*n^2 - 1.

Example

The first rows are:
   3
   5, 21, 15
   7, 55, 45, 39, 119, 77, 33, 65, 35

Prog

(PARI) M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1])

Crossrefs

See A321769 and A321770 for the other components.

See A322170 for the corresponding areas.

See A322181 for the corresponding perimeters.

Cf. A321782, A321783, A321784, A321785.

Cf. A046727.

Sequence in context: A007363 A103991 A224994 * A331395 A086175 A065926

Adjacent sequences: A321765 A321766 A321767 * A321769 A321770 A321771

Keyword

nonn,tabf

Author

Rémy Sigrist, Nov 18 2018

Status

approved